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Fractions Computations Operations

- Teaching for Conceptual Understanding

Introduction

- Represent the following operation using each of

the representations - Describe what the solution means in terms of each

representation you use. - Pictures
- Manipulatives
- Real World Situations
- Symbolic
- Oral/Written Language

Fractional Parts Counting

5 Fourths

3 Fourths

10 Fourths

10 Twelfths

Discussion

- What does the bottom number in a fraction tell

us? - What does the top number in a fraction tell us?

Misleading notion of fractions Top number tells

how many. Bottom number tells how many parts

to make a whole.

For example

- If a pizza is cut in 12 pieces, 2 pieces makes

1/6 of the pizza - Here, the bottom number does not tell us how many

parts make up the whole!

- Better Idea!
- We can assume the top number counts while the

bottom number tells what is being counted. - ¾ is a count of three things called fourths

Using this notion, what is another way to

say/write Thirteen sixths? Explain your rationale.

Activity

- Using the manipulatives, your task is to find a

single fraction that names the same amount as - must be able to provide an explanation for your

result. - Then, determine the mixed number for 17/4 and

provide a justification for your result.

Parts Whole Tasks

If 12 counters are ¾ of a set, how many counters

are in the full set?

If 10 counters are five-halves of a set, how many

counters are in one set? (What must be half of

one set?)

If purple is 1/3, what strip is the whole? If

dark green is 2/3, what strip is the whole? If

yellow is 5/4, what strip is 1 whole?

Purple

Dark Green

Yellow

Dark Green

If the dark green is the whole, what fraction is

the yellow strip?

Yellow

Dark Green

If the dark green strip is one whole, what

fraction is the blue strip?

Blue

Getting to Conceptual Understanding

- A father has left his three sons 35 camels to

divide among them in this way - One-half to one brother, one-third to another

brother, and one-ninth to the third brother. - How many camels does each brother receive?

Explain your solution to this problem.

Benchmarks of Zero, One-Half, and One

Sort the following fractions into three groups

close to zero, close to ½, and close to one.

Provide rationale and explanation for your

choices.

Close Fractions Name a fraction that is close to

one but not more than one. Name another fraction

that is even closer to one. Explain why you

believe this fraction is even closer to one than

the first. Show using manipulatives.

Partner/ Group Activity

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Exploring Ordering Fractions lt or gt

- Task Which fraction in each pair is greater?

Give an explanation for why you think so? - What are some of the usual approaches that

students would choose to use in this activity?

Why do you think this is?

Conceptual Thought Patterns

- More of the same size parts to compare 3/8 and

5/8 - - students will choose 5/8 as larger because 5gt3

right choice, wrong reason - - comparing 3/8 and 5/8 should be like comparing

3 apples and 5 apples

- Same number of parts, but parts of different

sizes - to compare ¾ and 3/7
- Misconception students will choose 3/7 as the

larger because 7 is more than 4 and the top

numbers are the same - However, if a whole is divided into 7 parts, the

parts will be smaller than if divided into only 4

parts thus, ¾ is larger - Like comparing 3 apples with 3 melons same

number of things but melons are bigger

- More or less than one-half or one whole
- comparing 3/7 and 5/8
- 3/7 is less than half of the number of sevenths

needed to make a whole, so 3/7 is less than

one-half - thus, 5/8 is more than one-half and is therefore

the larger fraction

- Distance from one-half or one whole
- compare 9/10 and ¾
- Misconception 9/10 is bigger because 9 and 10

are the bigger numbers - 9/10 is larger than ¾ because although each is

one fractional part away from a whole, tenths are

smaller than fourths and so 9/10 is closer to one

whole.

Addition and Subtraction Explorations

- Paul and his brother were each eating the same

kind of candy bar. Paul had ¾ of his candy bar.

His brother had 7/8 of a candy bar. How much

candy did the two boys have together? - Using drawings /or manipulatives, solve this

problem without setting it up in the usual manner

and finding common denominators? - Can you think of two different methods?

One possible Solution

We could take a fourth from the 7/8 and add it to

the ¾ to make a whole. That would leave 5/8.

Thus, the total eaten would be 1 5/8 of candy bar.

Using Cuisenaire Rods for Fraction Computation

- Jack and Jill ordered two identical sized pizzas,

one cheese and one pepperoni. Jack ate 5/6 of a

pizza and Jill ate ½ of a pizza. How much pizza

did they eat together? - What would we expect our students to show that

would demonstrate their conceptual understanding?

Solution

- Find a strip for the whole that allows both

fractions to be modeled.

Dark Green

Whole

Yellow

5/6

Light Green

1/2

Dark Green

Yellow

Light Green

Red

A Red is 1/3 of the Dark Green, so the solution

is 1 1/3.

Multiplication of Fractions Beginning Concepts

- Using visual representations/manipulatives, model

the following computation - There are 15 cars in Michaels toy car

collection. Two-thirds of the cars are red. How

many red cars does Michael have?

- Using visual representations/manipulatives, now

model these following problems - You have ¾ of a pizza left. If you give 1/3 of

the left-over pizza to your brother, how much of

the whole pizza will your brother get? - Someone ate 1/10 of the cake, leaving only 9/10.

If you eat 2/3 of the cake that is left, how much

of the whole cake have you eaten? - Gloria used 2 ½ tubes of blue paint to paint the

sky in her picture. Each tube holds 4/5 ounce of

paint. How many ounces of blue paint did Gloria

use?

When pieces must be subdivided into smaller unit

parts, the problems become more challenging

- Zack had 2/3 of the lawn left to cut. After lunch

he cut ¾ of the grass that he had left. How much

of the whole lawn did he cut after lunch? - Bill drank 1/5 of his pop before lunch. After

lunch he drank 2/3 of what was left. How much pop

did he drink after lunch?

Division of Fractions Beginning Concepts

- Think about the following problem
- Cassie has 5 ¼ yards of ribbon to make three

bows for birthday packages. How much ribbon

should she use for each bow if she wants to use

the same length of ribbon for each? - What types of solutions would we anticipate our

students to come up with? - How could we model the solution using

manipulatives/multiple representations? - How many different ways?

- In the following problem, the parts must be split

into smaller parts - Mark has 1 ¼ hours to finish his three

household chores. If he divides his time evenly,

how many hours can he give to each?

Questions/Discussion

- Inverse vs. Reciprocal
- 4/5 - representations
- Think of the many forms that even the symbolic

can be represented decimals, rates, ratios,

etc. - Other? More???

- Applying this Reasoning
- Solving Problems

- A A student is sorting into stacks a room full

of food donated by the school for the local food

bank. He sorted of it before lunch and then

sorted of the remainder before school ended.

What part (fraction) of all the food will be left

for him to sort after school?

1 3

3 4

- B Use manipulatives to solve this problem.

Mark ate half of the candies in a bag. Leila ate

2/3 of what was left. Now there are 11 candies in

the bag. How many were in the bag at the start? - C Bills Snow Plow can plow the snow off the

schools parking lot in 4 hours. Janes plowing

company can plow the same parking lot in just 3

hours. How long would it take Bill and Jane to

plow the schools parking lot together?

Think of the math content involved with this

problem Think of some Before activities that

could be used. What would the debrief look/sound

like in the classroom after the task was complete?

Teaching Through Problem Solving

- D Mrs. Get Fit teaches Math and Phys-ed. To

incorporate Math into the Phys-ed class, she

divided the class into eight groups. There are

three students in each group. The first person in

the group runs ¼ of a lap of the track, the

second person runs 1/6 of the track, and the

third person runs 1/3 of a lap of the track. How

many laps of the track are run in total by all

eight teams combined?